In this work, the Riemannian metric \(\mathcal {N}_{p,k}\)  is chosen as the sum of metrics on \(\textup {St}_{p,k}\) , \(\mathcal {S}^{++}_k\)  and \(\mathcal {D}^{++}_p\) .
The metric on \(\textup {St}_{p,k}\)  is the so-called canonical metric [1]}, which yields the simplest geometry of \(\textup {St}_{p,k}\) .
The metrics on \(\mathcal {S}^{++}_k\)  and \(\mathcal {D}^{++}_p\)  are the affine-invariant ones.
Hence, our chosen metric on \(\mathcal {N}_{p,k}\)  is
\(\langle \xi ,\eta \rangle ^{\mathcal {N}_{p,k}}_{\theta } =\operatorname{tr}({\xi }_{{V}}^{\top }({I}_p - \frac{1}{2}{V}{V}^{\top }){\eta }_{{V}})\\+ \operatorname{tr}({\Lambda }^{-1}{\xi }_{{\Lambda }}{\Lambda }^{-1}{\eta }_{{\Lambda }})+ \operatorname{tr}({\Psi }^{-2}{\xi }_{{\Psi }}{\eta }_{{\Psi }}).\) 
